Dimension-six operators of a real scalar field (${\varphi }^6,{\varphi }^3\square \varphi ,(\square \varphi {)}^2$)

In four dimensions, a real scalar field and a partial derivative each have mass dimension one. Seven scalar monomials of dimension six can be listed before integration by parts: ${\varphi }^6$, ${\varphi }^2(\partial \varphi {)}^2$, ${\varphi }^3\square \varphi$, $\varphi {\square }^2\varphi$, ${\partial }_{\mu }\varphi {\partial }^{\mu }\square \varphi$, $(\square \varphi {)}^2$, and $({\partial }_{\mu }{\partial }_{\nu }\varphi {)}^2$. They reduce to the three displayed representatives modulo total derivatives. Leading equations of motion reduce them further to a single new ${\varphi }^6$ interaction, together with shifts of the existing mass and quartic coupling constant.

At first order in the suppression of an effective operator, a factor of the leading Euler-Lagrange field equation defines a redundant operator removable by a local field redefinition. For ${\mathcal{L}}_0=(\partial \varphi {)}^2/2-m^2{\varphi }^2/2-\lambda {\varphi }^4/24$, one obtains ${\varphi }^3\square \varphi \doteq -m^2{\varphi }^4-\lambda {\varphi }^6/6$ and $(\square \varphi {)}^2\doteq m^4{\varphi }^2+m^2\lambda {\varphi }^4/3+{\lambda }^2{\varphi }^6/36$. Lower-dimensional parameter shifts must accompany this reduction when $m\ne 0$; it is not an identity for untransformed off shell correlation functions.

Two Lagrangian densities differing by a total derivative yield the same action under suitable boundary conditions. For example ${\partial }_{\mu }({\varphi }^3{\partial }^{\mu }\varphi )=3{\varphi }^2(\partial \varphi {)}^2+{\varphi }^3\square \varphi$. Likewise $\varphi {\square }^2\varphi \simeq -\partial \varphi \cdot \partial \square \varphi \simeq (\square \varphi {)}^2\simeq ({\partial }_{\mu }{\partial }_{\nu }\varphi {)}^2$. These are algebraic relations modulo boundary terms, not uses of the equation of motion.